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Unformatted text preview: Chapter 1 Signals and Systems Signals convey information. Systems transform signals. This book is about developing an under- standing of both. We gain this understanding by dissecting their structure and by examining their interpretation. For systems, we look at the relationship between the input and output signals (this relationship is a declarative description of the system) and the procedure for converting an input signal into an output signal (this procedure is an imperative description of the system). A sound is a signal. We leave the physics of sound to texts on physics, and instead, show how a sound can be usefully decomposed into components that themselves have meaning. A musical chord, for example, can be decomposed into a set of notes. An image is a signal. We do not discuss the biophysics of visual perception, but instead show that an image can be usefully decomposed. We can use such decomposition, for instance, to examine what it means for an image to be sharp or blurred, and thus to determine how to blur or sharpen an image. The decomposition of signals into sinusoids is considered in later chapters. Signals can be more abstract (less physical) than sound or images. They can be, for example, a sequence of commands or a list of names. We develop models for such signals and the systems that operate on them, such as a system that interprets a sequence of commands from a musician and produces a sound. One way to get a deeper understanding of a subject is to formalize it, to develop mathematical models. Such models admit manipulation with a level of confidence not achievable with less formal models. We know that if we follow the rules of mathematics, then a transformed model still relates strongly to the original model. There is a sense in which mathematical manipulation preserves “truth” in a way that is elusive with almost any other intellectual manipulation of a subject. We can leverage this truth-preservation to gain confidence in the design of a system, to extract hidden information from a signal, or simply to gain insight. Mathematically, we model both signals and systems as functions. A signal is a function that maps a domain, often time or space, into a range, often a physical measure such as air pressure or light intensity. A system is a function that maps signals from its domain—its input signals—into signals in its range—its output signals. The domain and the range are both sets of signals ( signal spaces ). 1 2 CHAPTER 1. SIGNALS AND SYSTEMS Thus, systems are functions that operate on functions. We use the mathematical language of sets and functions to make our models unambiguous, precise, and manipulable. This language has its own notation and rules, which are reviewed in appendix A ....
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- Spring '08